Suppose I have a partition of $\mathbb{R}^N$ $$ \mathbb{R}^N = \bigcup_{i \in \mathbb{R}} \mathcal{X}_i \qquad \qquad \text{with } \mathcal{X_i} \cap \mathcal{X}_j = \emptyset \text{ if } i \neq j $$ and that for each value of $i\in\mathbb{R}^N$ I have a measure $\pi_i: \mathcal{X}_i \to [0, +\infty)$. Since all together the measures $\{\pi_i\}_{i\in \mathbb{R}}$ can "measure" the whole of $\mathbb{R}^N$, I would like to gather them together into one measure $$ \pi: \mathbb{R}^N \to [0, +\infty) $$ defined as $$ \pi(A) = \sum_{i\in\mathbb{R}} \pi_i(A \cap \mathcal{X}_i) $$
Does this measure $\pi$ have a name?(e.g. mixture/union of measures) Essentially I would like to be able to write something like this $\pi = \bigotimes_{i\in\mathbb{R}} \pi_i$, is this doable?
You are asking about a "sum" over the uncountable index set $\mathbb{R}$. This is problematical. If you think of the plane as the infinite disjoint union of a pencil of parallel lines, you cannot recover the area of a set from the area (measure) of its intersection with each of the lines by summing. You need an integral of some kind to formalize Cavalieri's principle.
I am sure this is discussed in the literature but I don't see immediately how to look up a reference.
Edit
Searches for measurable foliation and measurable fibration lead to some links that may be helpful (and many that are probably more technical that what you need).